Time-Lock Puzzles (TLPs) have been developed to securely transmit sensitive information into the future without relying on a trusted third party. Multi-instance TLP is a scalable variant of TLP that enables a server to efficiently find solutions to different puzzles provided by a client at once. Nevertheless, existing multi-instance TLPs lack support for (verifiable) homomorphic computation. To address this limitation, we introduce the "Multi-Instance partially Homomorphic TLP" (MH-TLP), a multi-instance TLP supporting efficient verifiable homomorphic linear combinations of puzzles belonging to a client. It ensures anyone can verify the correctness of computations and solutions. Building on MH-TLP, we further propose the "Multi-instance Multi-client verifiable partially Homomorphic TLP" (MMH-TLP). It not only supports all the features of MH-TLP but also allows for verifiable homomorphic linear combinations of puzzles from different clients. Our schemes refrain from using asymmetric-key cryptography for verification and, unlike most homomorphic TLPs, do not require a trusted third party. A comprehensive cost analysis demonstrates that our schemes scale linearly with the number of clients and puzzles.

翻译：时间锁谜题（TLP）旨在无需依赖可信第三方的情况下，将敏感信息安全地传递至未来。多实例TLP作为TLP的可扩展变体，使服务器能够一次性高效求解客户端提供的不同谜题。然而，现有多实例TLP缺乏对（可验证）同态计算的支持。为突破此限制，本文提出"多实例部分同态TLP"（MH-TLP），该方案支持对同一客户端的谜题进行高效可验证的同态线性组合，并确保任何参与者均可验证计算过程与解的正确性。基于MH-TLP，我们进一步提出"多实例多客户端可验证部分同态TLP"（MMH-TLP），该方案不仅具备MH-TLP的全部特性，还支持跨客户端谜题的可验证同态线性组合。我们的方案避免使用非对称密码学进行验证，且与多数同态TLP不同，无需依赖可信第三方。全面的成本分析表明，我们的方案在客户端数量与谜题数量上均呈现线性增长。